derived under the assumption of zero torsion. In [1, p.81, (3.88)] and [2, p.128, (3.140)] we find the remark:

For a general connection there would be additional terms involving the torsion tensor.

However, M.W. Evans knows more: In his GCUFT book [3, p.325,(D.8)]and in [3a, (D.8)] we find:

                D_ρR^α_σμν + D_μR^α_σνρ + D_νR^α_σρμ = 0 .                                 (D.8)

. . . The second Bianchi identity is true for ANY gamma connection.

Who is right?

                DÙR^a_b = dÙR^a_b + ω^a_c ÙR^c_b + ω^c_b ÙR^a_c = 0                     (D.1)

which can be found also in [1, (3.141)] and [2, (J.32)] and can hence be considered as confirmed. However, the following equation

                D_ρR^a_bμν = ∂_ρR^a_bμν + ω^a_ρc R^c_bμν + ω^c_ρb R^a_cμν                 (D.3)
                                et cyclicum.

has evidently missing terms since due to the rules of covariant differentiation each index should produce an additional term besides the partial derivative, but there exist no terms corresponding to the indices μ and ν. The sum

(3)                 − Γ_ρ^α_μ R^a_bαν − Γ_ρ^α_ν R^a_bμα

is missing which under cyclical summation most likely gives rise to terms containing the torsion tensor.

Thus, Evans' 2nd Bianchi identity (D.8) is invalid in case of non-vanishing torsion.

Evans in [3, p.302], [3a, (D.8)]:

In contrast, note carefully that the second Bianchi identity (17.4) (= (D.8)) is ALWAYS true for any type of connection, because it is fundamentally the cyclic sum of commutators of covariant derivatives [2]:

                [[D_λ,D_ρ],D_σ] + [[D_ρ,D_σ],D_λ] + [[D_σ,D_λ],D_ρ] := 0.                 (17.9)/(D.9)

Equ. (17.9) is true but NOT so its application to R^a_bμν as performed by (D.3) and yielding the invalid eqns. (D.8) and (17.4).